For a rigid space \(X/K\) , let \(X_v\)  denote the \(v\) -site of \(X\)  from [1]}. Let \({\mathbf {Z}_\ell } = \lim _n \underline{\mathbf {Z}/\ell ^n}\)  be the displayed inverse limit of constant sheaves, regarded as a sheaf of (abstract) rings. Any perfect complex \(M \in D(\mathbf {Z}_\ell )\)  yields a “constant” \(\ell \) -complete sheaf \(\underline{M} := \lim _n \underline{M/\ell ^n}\in D(X_v, \mathbf {Z}_\ell )\) . Our goal is to build a theory of Zariski-constructible \(\mathbf {Z}_\ell \) -complexes on a rigid space \(X\)  where the locally constant objects are twisted forms of \(\underline{M}\)  for \(M \in D_{perf}(\mathbf {Z}_\ell )\) . We first recall the basic notion of “étale \(\mathbf {Z}_\ell \) -sheaves” that is introduced in [1]} for the purposes of defining operations.
